Finite element method for singularly perturbed problems with two parameters on a Bakhvalov-type mesh in 2D

On a Bakhvalov-type mesh widely used for boundary layers, we consider the finite element method singularly perturbed elliptic problems with two parameters on unit square. It is very challenging task to analyze uniform convergence of this in 2D. The existing analysis tool, quasi-interpolation, only applicable one-dimensional case because complexity In paper, powerful Lagrange-type interpolation, proposed, which simple and effective can be both 1D application interpolation 2D must handled carefully. Some correction terms introduced maintain homogeneous Dirichlet condition. These are difficult traditional do not work them. To overcome difficulty, derive delicate estimation width some mesh. Moreover, adopt different strategies layers. Finally, prove optimal order. Numerical results verify theoretical analysis.